Green Function Solution Method

Introduction

The use of the Free-Surface Green Function to solve the Standard Linear Wave Scattering Problem has proved one of the most powerful methods, primarily because of its very general nature so that it can deal with complicated boundary conditions. It also solves explicity for the boundary conditions at infinite (Sommerfeld Radiation Condition)

Standard Linear Wave Scattering Problem

The equations are the following

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math])

[math]\displaystyle{ \partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B, }[/math]

where [math]\displaystyle{ \mathcal{L} }[/math] is a linear operator which relates the normal and potential on the body surface through the physics of the body.

We begin with the Standard Linear Wave Scattering Problem

[math]\displaystyle{ \nabla^{2}\phi=0, \, -\infty\lt z\lt 0,\,\,\,\mathbf{x}\notin \Omega }[/math] [math]\displaystyle{ \frac{\partial\phi}{\partial z}=0, \, z=h, }[/math] [math]\displaystyle{ \frac{\partial\phi}{\partial z} = k_{\infty}\phi,\,z=0,\,\,\mathbf{x}\notin\Omega, }[/math] [math]\displaystyle{ \frac{\partial\phi}{\partial z} = L\phi, \, z\in\partial\Omega, }[/math]

We then use Green's second identity If φ and ψ are both twice continuously differentiable on U, then

[math]\displaystyle{ \int_U \left( \psi \nabla^2 \varphi - \varphi \nabla^2 \psi\right)\, dV = \oint_{\partial U} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS }[/math]

If we then substitiute the Free-Surface Green Function which satisfies the following equations (plus the Sommerfeld Radiation Condition far from the body)

[math]\displaystyle{ \nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\infty\lt z\lt 0 }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z}=0, \, z=-h, }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z} = k_{\infty}\phi,\,z=0. }[/math]

for ψ we obtain

[math]\displaystyle{ \phi^{in} + \int_{\partial \Omega }\left( G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x} ^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi _{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime } = \left( \begin{matrix} 0, \,\,\,x\notin U \cup \partial U, \\ \phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial U, \\ \phi(\mathbf{x}),\,\,\,\mathbf{x} \in U, \end{matrix} \right. }[/math]